name____________________ period _______ lab partners___________________________________
Simple Harmonic Motion
Introduction
Lots of things vibrate or oscillate. A vibrating tuning fork, a moving child’s playground swing, and the
loudspeaker in a radio are all examples of physical vibrations. There are also electrical and acoustical
vibrations, such as radio signals and the sound you get when blowing across the top of an open bottle.
One simple system that vibrates is a mass hanging from a spring--the focus of this lab (see Figure #1.)
The force F applied by an ideal spring is proportional to how much it is stretched or compressed x, that is F =
-kx (equation #1). Given this force behavior, the up and down motion of the mass is called simple
harmonic and the vertical position can be modeled with (equation #2)
In this equation, y is the vertical displacement from the equilibrium position, A is the amplitude of the
motion, f is the frequency of the oscillation, t is the time, and  is a phase constant.
Recall the frequency f is measured in cycles per second; the reciprocal of that 1 / f = period T has units of
seconds per cycle. Not surprisingly, f = 1 / T (equation #3). This experiment will clarify these terms.
Objectives

Measure the position and velocity as a function of time for an oscillating mass and spring system.
 Determine the amplitude, period, frequency and phase constant of the observed simple harmonic
motion, and better appreciate the meaning of these terms.
 Fit the data for motion of mass / spring system to mathematical model of simple harmonic motion.
Materials
computer
Vernier computer interface
Logger Pro
Vernier Motion Detector
200 g and 300 g masses
ring stand, rod, and clamp
spring, with k ~10 N/m
twist ties
Figure #1
wire basket
Procedure
1. Attach the spring to a horizontal rod connected to the ring stand and hang mass from the spring (Figure 1.)
Securely fasten the 200 g mass to the spring and spring to rod, using twist ties so the mass cannot fall.
2. Connect the Motion Detector to the DIG/SONIC 1 channel of the interface. If the Motion Detector has a
switch, set it to Normal. Place the Motion Detector at least 75 cm below the mass. Make sure there are no
objects between the detector and mass, such as a table edge. Place wire basket over the Motion Detector
3. Open Logger Pro. Open the file “15 Simple Harmonic Motion” from the Physics with Vernier folder.
4. Make a preliminary run to make sure things are set up correctly. Lift the mass upward a few centimeters
and release. The mass should oscillate along a vertical line only. Click
to begin data collection.
After 10 s, data collection will stop. The position graph should show a clean sinusoidal curve. If it has flat
regions or spikes, reposition the Motion Detector and try again.
5. Measure the equilibrium position of the 200 g mass. Do this by allowing the mass to hang free and at rest.
Click
to begin data collection. After collection stops, click the Statistics button, , to determine the
average distance from the detector. Record this distance (y0) in your data table.
6. Now lift the mass upward about 5 cm and release it. The mass should oscillate along a vertical line only.
Click
to collect data. Examine the graphs (this is Run #1). The pattern you are observing is
characteristic of simple harmonic motion (that is, graphs of position and velocity are sine curves).
7. Using the position graph, measure the time interval between maximum positions. This is the period, T, of
the motion. Based on your period measurement, calculate the frequency using equation #3. Record the period
and frequency of this motion in Data Table #1, and show sample calculations in the space provided there.
8. The amplitude, A, of simple harmonic motion is the maximum distance from the equilibrium position.
Estimate values for the amplitude from your position graph. Enter the values in your data table. If you drag
the mouse from one peak to another you can read the Δx or dx time interval.
9. Run #2: repeat Steps 5--8 with the same 200 g mass, starting with a larger amplitude than in Run #1.
10. Run #3: change the mass to 300 g and repeat Steps 5–8. Use an amplitude of about 5 cm. Keep the graph
/ data for this run on the screen. You will need it for the Data Handling / Analysis below.
Data Table #1
Run
Mass
(g)
y0
(cm)
A
(cm)
T
(s)
f
(Hz)
Show Sample Calculations in space below:
1
2
3
Data Handling / Analysis / Questions -- Do these on separate sheet, followed by your conclusion.
1. View the graphs of the last run on the screen. Compare the position vs. time and the velocity vs. time
graphs. How are they the same? How are they different?
2. Turn on the Examine mode by clicking the Examine button, . Move the mouse cursor back and forth
across the graph to view the data values for the last run on the screen. Where is the mass when the
velocity is zero? Where is the mass when the velocity is greatest?
3. Does the frequency, f, appear to depend on the amplitude of the motion? Do you have enough data to
draw a firm conclusion?
4. Does the frequency, f, appear to depend on the mass used? Did it change much in your tests?
5. You can compare your experimental data to the sinusoidal function model using the Manual Curve Fit
feature of Logger Pro. Try it with your Run #3 data. (You'll follow instructions presented in the box
below.) The model equation presented as equation #2 gives the displacement from equilibrium. However,
your Motion Detector reports the distance from the detector. To compare the model to your data, add the
equilibrium distance to the model; that is, use
y  A sin 2ft     y0 equation #4 where y0 represents the equilibrium distance.
Instructions for Analysis Using Logger Pro's Manual Curve Fit
For This Lab's Run #3
1) Click once on the position graph to select it.
2) Choose Curve Fit from the Analyze menu.
3) Select Manual as the Fit Type.
4) Select the Sine function from the General Equation list.
5) The Sine equation is of the form y=A*sin(Bt +C) + D. Compare this to
the form of the equation above (equation #4) to match variables; e.g., φ
corresponds to C, and 2πf corresponds to B.
6) Adjust the values for A, B and D to reflect your values for A, f and y0.
You can either enter the values directly in the dialog box or you can use the
up and down arrows to adjust the values.
7) The phase parameter φ is called the phase constant and is used to adjust
the y value reported by the model at t = 0 so that it matches your data. Since
data collection did not necessarily begin when the mass was at the
equilibrium position, φ is needed to achieve a good match.
8) The optimum value for φ will be between 0 and 2π = 6.28. Find a value
for φ that makes the model come as close as possible to data for Run #3.
You can do this by trial and error. Realize that the RMSE value measures
how good the fit it. RMSE stands for "root mean square error." The
smaller it is, the better the fit between your model and the actual
graph of the data. You may also want to ad just y0, A, and f to improve
the fit. Record results in Data Table #2.
9) Write down the equation that best matches your data. Include RMSE.
6. Does the model fit the data well? How can you tell?
7.
Click
, and optionally print your graph.
Data Table #2
Documenting Fitting a Sine Curve to your Data
trial #
A
B
C
D
RMSE
1
2
3
4
5
6
7
note: Ideally, your RMSE value will get smaller as trials continue!
Final Best Fit Equation: _____________________
RMSE value: ______